Consumption ECON 30020: Intermediate

Prof. Eric Sims

University of Notre Dame

Fall 2016

1 / 36 of Macro

I We now move from the long run (decades and longer) to the medium run (several years) and short run (months up to several years)

I In long run, we did not explicitly model most economic decision-making – just assumed rules (e.g. consume a constant fraction of income)

I Building blocks of the remainder of the course are decision rules of optimizing agents and a concept of equilibrium

I Will be studying optimal decision rules first I Framework is dynamic but only two periods (t, the present, and t + 1, the future)

I Consider representative agents: one household and one firm I Unrealistic but useful abstraction and can be motivated in world with heterogeneity through insurance markets

2 / 36 Consumption

I Consumption the largest expenditure category in GDP (60-70 percent)

I Study problem of representative household I Household receives exogenous amount of income in periods t and t + 1

I Must decide how to divide its income in t between consumption and /borrowing

I Everything real – think about one good as “fruit”

3 / 36 Basics

I Representative household earns income of Yt and Yt+1. Future income known with certainty

I Consumes Ct and Ct+1

I Begins life with no , and can save St = Yt − Ct (can be negative, which is borrowing)

I Earns/pays real rt on saving/borrowing

I Household a price-taker: takes rt as given I Do not model a financial intermediary (i.e. bank), but assume existence of option to borrow/save through this intermediary

4 / 36 Budget Constraints

I Two flow budget constraints in each period:

Ct + St ≤ Yt

Ct+1 + St+1 − St ≤ Yt+1 + rt St

I Saving vs. : saving is a flow and savings is a stock. Saving is the change in the stock

I As written, St and St+1 are stocks I In period t, no distinction between stock and flow because no initial stock

I St+1 − St is flow saving in period t + 1; St is the stock of savings household takes from t to t + 1, and St+1 is the stock it takes from t + 1 to t + 2

I rt St : income earned on the stock of savings brought into t + 1

5 / 36 Terminal Condition and the IBC

I Household would not want St+1 > 0. Why? There is no t + 2. Don’t want to die with positive assets

I Household would like St+1 < 0 – die in debt. Lender would not allow that

I Hence, St+1 = 0 is a terminal condition (sometimes “no Ponzi”)

I Assume budget constraints hold with equality (otherwise leaving income on the table), and eliminate St , leaving:

Ct+1 Yt+1 Ct + = Yt + 1 + rt 1 + rt

I This is called the intertemporal budget constraint (IBC). Says that present discounted value of stream of consumption equals present discounted value of stream of income.

6 / 36 Preferences

I Household gets from how much it consumes

I Utility function: u(Ct ). “Maps” consumption into utils 0 I Assume: u (Ct ) > 0 (positive ) and 00 u (Ct ) < 0 (diminishing marginal utility) I “More is better, but at a decreasing rate” I Example utility function:

u(Ct ) = ln Ct 0 1 u (Ct ) = > 0 Ct 00 −2 u (Ct ) = −Ct < 0

I Utility is completely ordinal – no meaning to magnitude of utility (it can be negative). Only useful to compare alternatives

7 / 36 Lifetime Utility

I Lifetime utility is a weighted sum of utility from period t and t + 1 consumption:

U = u(Ct ) + βu(Ct+1)

I 0 < β < 1 is the discount factor – it is a measure of how impatient the household is.

8 / 36 Household Problem

I Technically, household chooses Ct and St in first period. This effectively determines Ct+1

I Think instead about choosing Ct and Ct+1 in period t

max U = u(Ct ) + βu(Ct+1) Ct ,Ct+1 s.t. Ct+1 Yt+1 Ct + = Yt + 1 + rt 1 + rt

9 / 36 Euler Equation

I First order optimality condition is famous in – the “Euler equation” (pronounced “oiler”)

0 0 u (Ct ) = β(1 + rt )u (Ct+1)

I Intuition and example with log utility I Necessary but not sufficient for optimality I Doesn’t determine level of consumption. To do that need to combine with IBC

10 / 36 Indifference Curve

I Think of Ct and Ct+1 as different (different in time dimension)

I Indifference curve: combinations of Ct and Ct+1 yielding fixed overall level of lifetime utility

I Different indifference curve for each different level of lifetime utility. Direction of increasing preference is northeast

I Slope of indifference curve at a point is the negative ratio of marginal :

0 u (Ct ) slope = − 0 βu (Ct+1)

00 I Given assumption of u (·) < 0, steep near origin and flat away from it

11 / 36 Budget Line

I Graphical representation of IBC

I Shows combinations of Ct and Ct+1 consistent with IBC holding, given Yt , Yt+1, and rt I Points inside budget line: do not exhaust resources I Points outside budget line: infeasible

I By construction, must pass through point Ct = Yt and Ct+1 = Yt+1 (“endowment point”) I Slope of budget line is negative gross real interest rate:

slope = −(1 + rt )

12 / 36 Optimality Graphically

I Objective is to choose a consumption bundle on highest possible indifference curve I At this point, indifference curve and budget line are tangent (which is same condition as Euler equation)

𝐶𝐶𝑡𝑡+1

(1 + ) +

𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1

, (2)

𝐶𝐶2 𝑡𝑡+1 𝑌𝑌𝑡𝑡+1 , (3) = 𝐶𝐶3 𝑡𝑡, +1 𝑈𝑈 𝑈𝑈2 𝐶𝐶0 𝑡𝑡+1 , = (0) 1 𝑡𝑡+1 𝐶𝐶 (1) = 𝑈𝑈 𝑈𝑈1

𝑈𝑈 𝑈𝑈0 , , , , + 𝐶𝐶𝑡𝑡 1 +𝑡𝑡+1 13 / 36 𝑌𝑌𝑡𝑡 𝐶𝐶0 𝑡𝑡 𝐶𝐶3 𝑡𝑡 𝐶𝐶2 𝑡𝑡 1 𝑡𝑡 𝑌𝑌 𝐶𝐶 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡

I What we want is a decision rule that determines Ct as a function of things which the household takes as given – Yt , Yt+1, and rt I Consumption function:

d Ct = C (Yt , Yt+1, rt )

I Can use indifference curve - budget line diagram to qualitatively figure out how changes in Yt , Yt+1, and rt affect Ct

14 / 36 Increases in Yt and Yt+1

I An increase in Yt or Yt+1 causes the budget line to shift out horizontally to the right

I In new optimum, household will locate on a higher indifference curve with higher Ct and Ct+1 I Important result: wants to increase consumption in both periods when income increases in either period

I Wants its consumption to be “smooth” relative to its income I Achieves smoothing its consumption relative to income by adjusting saving behavior: increases St when Yt goes up, reduces St when Yt+1 goes up ∂C d ∂C d I Can conclude that > 0 and > 0 ∂Yt ∂Yt+1 ∂C d I Further, < 1. Call this the marginal propensity to ∂Yt consume, MPC

15 / 36 Increase in rt

I A little trickier I Causes budget line to become steeper, pivoting through endowment point I Competing income and substitution effects: I Substitution effect: how would consumption bundle change when rt increases and income is adjusted so that household would locate on unchanged indifference curve? I Income effect: how does change in rt allow household to locate on a higher/lower indifference curve?

I Substitution effect always to reduce Ct , increase St I Income effect depends on whether initially a borrower (Ct > Yt , income effect to reduce Ct ) or saver (Ct < Yt , income effect to increase Ct )

16 / 36 Borrower

𝐶𝐶𝑡𝑡+1

Hypothetical bundle with new on same indifference curve 𝑟𝑟𝑡𝑡

𝑌𝑌𝑡𝑡,+1 ℎ, 𝐶𝐶0 𝑡𝑡+1 1 𝑡𝑡+1 𝐶𝐶 , Original bundle

0 𝑡𝑡+1 𝐶𝐶 New bundle

, , , 𝐶𝐶𝑡𝑡 ℎ 𝑡𝑡 1 𝑡𝑡 0 𝑡𝑡 𝑌𝑌 𝐶𝐶 𝐶𝐶0 𝑡𝑡 𝐶𝐶

I Sub effect: ↓ Ct . Income effect: ↓ Ct I Total effect: ↓ Ct

17 / 36 Saver

𝐶𝐶𝑡𝑡+1 New bundle

, Hypothetical bundle with new on same indifference curve 𝐶𝐶1 𝑡𝑡+1 𝑡𝑡 , 𝑟𝑟 ℎ Original bundle 0 𝑡𝑡+1 𝐶𝐶 ,

𝐶𝐶0 𝑡𝑡+1

𝑌𝑌𝑡𝑡+1

, , 𝑡𝑡 ℎ 𝐶𝐶 0 𝑡𝑡 0 𝑡𝑡 𝐶𝐶 ,𝐶𝐶 𝑌𝑌𝑡𝑡 𝐶𝐶1 𝑡𝑡

I Sub effect: ↓ Ct . Income effect: ↑ Ct I Total effect: ambiguous

18 / 36 The Consumption Function

I We will assume that the substitution effect always dominates for the interest rate

I Qualitative consumption function (with signs of partial derivatives) Ct = C (Yt , Yt+1, rt ). + + −

I Technically, partial derivative itself is a function I However, we will mostly treat the partial with respect to first argument as a parameter we call the MPC

19 / 36 Algebraic Example with Log Utility

I Suppose u(Ct ) = ln Ct I Euler equation is:

Ct+1 = β(1 + rt )Ct

I Consumption function is:   1 Yt+1 Ct = Yt + 1 + β 1 + rt

1 I MPC: 1+β . Go through other partials

20 / 36 Permanent Income Hypothesis (PIH)

I Our analysis consistent with Friedman (1957) and the PIH I Consumption ought to be a function of “permanent income” I Permanent income: present value of lifetime income

I Special case: rt = 0 and β = 1: consumption equal to average lifetime income I Implications: 1. Consumption forward-looking. Consumption should not react to changes in income that were predictable in the past 2. MPC less than 1 3. Longer you live, the lower is the MPC

I Important empirical implications for econometric practice of the day. Regression of Ct on Yt will not identify MPC (which is relevant for things like fiscal multiplier) if in historical data changes in Yt are persistent

21 / 36 Applications and Extensions

I We will consider several applications / extensions: 1. Wealth 2. Permanent vs. transitory changes in income 3. Uncertainty 4. Random walk

22 / 36 Wealth

I Allow household to begin life with stock of wealth Ht−1. Real price of this asset in t is Qt I Household can accumulate more of this asset or sell it I Period t constraint:

Ct + St + Qt (Ht − Ht−1) ≤ Yt

I Period t + 1 constraint:

Ct+1 + St+1 + Qt+1(Ht+1 − Ht ) ≤ Yt + (1 + rt )St

I Imposing terminal conditions, IBC is:

Ct+1 Yt+1 Qt+1Ht Ct + + Qt Ht = Yt + + Qt Ht−1 + 1 + rt 1 + rt 1 + rt

23 / 36 Simplifying Assumptions and the Consumption Function

I First, assume household must choose Ht = 0. It simply sells off the asset in period t at price Qt :

Ct+1 Yt+1 Ct + = Yt + + Qt Ht−1 1 + rt 1 + rt

I Increase in Qt then functions just like increase in Yt

d Ct = C (Yt , Yt+1, rt , Qt ) + + − +

I Empirical application: stock boom of 1990s (increase in Qt )

24 / 36 Alternative Simplifying Assumption

I Assume Ht−1 = 0, and assume that household must purchase an exogenous amount of the asset, Ht (e.g. has to buy a house)

I IBC:   Ct+1 Yt+1 Qt+1 Ct + = Yt + + Ht − Qt 1 + rt 1 + rt 1 + rt

I Increase in Qt+1: functions like increase in Yt+1:

d Ct = C (Yt , Yt+1, rt , Qt , Qt+1) + + − − +

I Empirical applications: house price boom of 2000s (anticipated increase in Qt+1)

25 / 36 Permanent vs. Transitory Changes in Income

I Go back to standard consumption function:

d Ct = C (Yt , Yt+1, rt )

I Take total derivative (differs from partial derivative in allowing everything to change):

∂C d (·) ∂C d (·) ∂C d (·) dCt = dYt + dYt+1 + drt ∂Yt ∂Yt+1 ∂rt

dC ∂C d (·) I If just dY 6= 0, then t is equal to partial t dYt ∂Yt I But if changes in income are persistent (i.e. dYt > 0 ⇒ d dY > 0), then dCt > ∂C (·) t+1 dYt ∂Yt I Implication: consumption reacts more to a change in income the more persistent is that change in income

26 / 36 Application: Tax Cuts

I Suppose household pays taxes, Tt and Tt+1, to government each period, so net income is Yt − Tt and Yt+1 − Tt+1. Consumption function is:

d Ct = C (Yt − Tt , Yt+1 − Tt+1, rt )

I A cut in taxes is equivalent to an increase in income I Implication: tax cuts will have bigger stimulative effects on consumption the more persistent the tax cuts are I Empirical studies: Shaprio and Slemrod (2003) and Shapiro and Slemrod (2009)

I Initial installment of Bush tax cuts in 2001 was close to permanent (ten years). Theory predicts consumption ought to react a lot. It didn’t. I Tax rebates 2008: known to be only one time. Theory predicts consumption should react comparatively little. It did.

27 / 36 Uncertainty

I Suppose that future income is not known with certainty h I Say it can take on two values: Yt+1 with probability p, and l Yt+1 with probability 1 − p. Expected future income is:

h l E (Yt+1) = pYt+1 + (1 − p)Yt+1

I Household will want to maximize expected lifetime utility. Utility from future consumption is not known, because future consumption isn’t known given uncertainty about income I Euler equation looks the same, but has an expectation operator: 0  0  u (Ct ) = β(1 + rt )E u (Ct+1)

I Key insight: expected value of a non-linear function is not equal to the function of the expected value.

28 / 36 Expected Marginal Utility vs. Marginal Utility of Expected Consumption

000 0 0 I Assume u (·) > 0. Then E [u (Ct+1)] > u (E [Ct+1])

( ) ′ 𝑢𝑢 𝐶𝐶𝑡𝑡+1

( ) ′ 𝑙𝑙 𝑢𝑢 𝐶𝐶𝑡𝑡+1

[ ( )] ′ 𝐸𝐸 𝑢𝑢 𝐶𝐶𝑡𝑡+1 ( [ ]) ′ 𝑢𝑢 𝐸𝐸 𝐶𝐶𝑡𝑡+1 ( ) ′ ℎ 𝑢𝑢 𝐶𝐶𝑡𝑡+1

[ ] 𝑡𝑡+1 𝑙𝑙 ℎ 𝐶𝐶 𝐶𝐶𝑡𝑡+1 𝐸𝐸 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡+1 29 / 36 Increase in Uncertainty: Mean-Preserving Spread

0 I Raises E [u (Ct+1)]

( ) ′ 𝑢𝑢 𝐶𝐶𝑡𝑡+1

uncertainty [ ( )] ( ) ′ , ↑ → ↑ 𝐸𝐸 𝑢𝑢 𝐶𝐶𝑡𝑡+1 ′ 𝑙𝑙 𝑢𝑢 𝐶𝐶1 𝑡𝑡+1

( , ) ′ 𝑙𝑙 0 𝑡𝑡+1 [𝑢𝑢 (𝐶𝐶 , )] ′ 𝐸𝐸[𝑢𝑢 (𝐶𝐶1,𝑡𝑡+1 )] ( ′ [ ]) 0 𝑡𝑡+1 𝐸𝐸′ 𝑢𝑢 𝐶𝐶 𝑢𝑢 𝐸𝐸 𝐶𝐶𝑡𝑡+1

( , ) ′ ( ℎ ) 𝑢𝑢 𝐶𝐶0,𝑡𝑡+1 ′ ℎ 𝑢𝑢 𝐶𝐶1 𝑡𝑡+1

, , [ ] , , 𝑡𝑡+1 𝑙𝑙 𝑙𝑙 ℎ ℎ 𝐶𝐶 𝐶𝐶1 𝑡𝑡+1 𝐶𝐶0 𝑡𝑡+1 𝐸𝐸 𝐶𝐶𝑡𝑡+1 𝐶𝐶0 𝑡𝑡+1 𝐶𝐶1 𝑡𝑡+1

30 / 36 Precautionary Saving

I Increase in uncertainty raises expected marginal utility of 0 consumption E [u (Ct+1)] I For the Euler equation to hold, need to adjust current 0 consumption to raise current marginal utility, u (Ct ) I This requires increasing saving – consume less holding current income fixed

I Intuition: bad state of the world hurts you more than the good state helps you, so you adjust behavior in present to effectively “self-insure” against bad future state

I Empirical application: high uncertainty and weak consumption during and in wake of Great Recession

31 / 36 Random Walk Hypothesis

I Suppose that β(1 + r) = 1 0 0 I Euler equation is then implies u (Ct ) = E [u (Ct+1)] 000 I Suppose that u (·) = 0 (so no precautionary saving). Then this implies that E [Ct+1] = Ct I In expectation, future consumption ought to equal current consumption. This is the “random walk” hypothesis

I Doesn’t mean that future consumption always equals current consumption

I But it does imply future changes in consumption ought to not be predictable, because in expectation future consumption should equal current consumption

I Random walk model due to Hall (1978)

32 / 36 Empirical Tests

I Random walk hypothesis one of the most tested macroeconomic theories I Generally fails: I Parker (1999): exploits facts about social security withholding and predictable changes over course of year. Consumption reacts to predictable changes in take home pay I Evans and Moore (2012): look at relationship between receipt of paycheck (which is predictable) and within-month mortality cycle

33 / 36 Borrowing Constraints

I Empirical failures can potentially be accounted for by borrowing constraints I Simplest form of a borrowing constraint: you can’t. St ≥ 0. Introduces kink into budget line

𝐶𝐶𝑡𝑡+1

(1 + ) +

𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1

𝑌𝑌𝑡𝑡+1 Infeasible if 0

𝑆𝑆𝑡𝑡 ≥

+ 𝐶𝐶𝑡𝑡 1 +𝑡𝑡+1 𝑌𝑌𝑡𝑡 𝑌𝑌 34 / 36 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 Binding Borrowing Constraint

I If borrowing constraint “binds” you locate at the kink in the budget line (i.e. Euler equation does not hold)

𝐶𝐶𝑡𝑡+1

, = ,

𝐶𝐶0 𝑡𝑡+1 𝑌𝑌0 𝑡𝑡+1

, ,

𝐶𝐶0 𝑑𝑑 𝑡𝑡+1

= , , , , 𝐶𝐶𝑡𝑡 𝐶𝐶0 𝑡𝑡 𝑌𝑌0 𝑡𝑡 𝐶𝐶0 𝑑𝑑 𝑡𝑡 35 / 36 Implications of a Binding Borrowing Constraint

I Current consumption equals current income I Means that if household gets more income, will spend all of it I Further means that if household expects more income in future, can’t adjust consumption until the future – consumption will react to anticipated change in income

I Potential resolution of some empirical failures of random walk / PIH model

I Also has policy implications. Makes sense to target taxes/transfers to households likely to be borrowing constrained if objective is to stimulate consumption

36 / 36